Institute of Metals Division -Measurement of Approximately Cylindrical Particles in Opaque Samples

Relationships are derived between average dimensions measured on a polished cross section and the spatial dimensions of particles dispersed as uniform cylinders. The equations are applicable to the measurement of rods or plates without the stipulation that they be extremely thin. QUANTITATIVE analysis of the elements of mi-crostructures has been limited by a lack of knowledge concerning the relationships between observations on plane surfaces and the spatial nature of the structure. This paper is an extension of previous efforts to establish equations suitable for the quantitative investigation of microstructures. The equations derived fill two gaps in the range of particle shapes for which measurements may be required, namely those shapes lying between extremely thin plates and equiaxed particles (approximately spherical), and those intermediate between equiaxed particles and extremely thin rods or needles. In a previous paper' equations were developed relating dimensions measured on a polished cross section to the average spatial dimensions of particles in the form of spheres, extremely thin plates, and extremely thin rods. For extremely thin rods the dimensions measured on a polished cross section were found to depend on the radius only, so that no information concerning the rod lengths or number of rods per unit volume could be obtained from observations on a cross section. In this paper equations are derived relating dimensions measured on a polished cross section to the spatial dimensions of particles in the form of identical cylinders. These equations may be used to evaluate the dimensions of rods, as well as the dimensions of plates, without the stipulation that the plates or rods be extremely thin. The impossibility of evaluating the length of extremely thin rods from planar measurements appears as an increase in the amount of data required for a given precision, as the length-radius ratio increases. Consider a sample containing particles of a phase in the form of cylinders of uniform radius T and height t. If the cylinders are randomly oriented, as in a sufficiently large sample of a fine grained poly-crystalline material, area and lineal analysis may be carried out with parallel cross-sectional planes and lineal traverses. If the cylinders are not randomly oriented, it is necessary to randomize the orientations of the cross-sectional planes and traverse lines. Let a unit cube be cut from the sample, and a cross-section plane be passed through the cube parallel to one of the cube faces. The number of particles N, cut by the cross-sectional plane per unit area is equal to the number of particles N, per unit volume times the probability p1 of a plane intersecting a single randomly positioned and randomly oriented cylinder in the cube. Let J be the maximum cylinder dimension in the direction normal to the cross-section planes, as shown in Fig. 1. Since, of the various possible positions for the cross-sectional plane over the unit length from top to bottom of the cube, only those positions existing over the length J